Interval and fuzzy physics-informed neural networks for uncertain fields

TL;DR

This paper introduces interval and fuzzy physics-informed neural networks (iPINNs and fPINNs) to solve PDEs with uncertain parameters, avoiding traditional sampling methods and requiring no correlation length knowledge.

Contribution

The work develops novel PINN-based methods for interval and fuzzy PDEs, providing bounded solutions without reliance on correlation data or Monte Carlo simulations.

Findings

iPINNs and fPINNs effectively solve uncertain PDEs with bounded solutions.

The methods eliminate the need for correlation length specifications.

They retain all advantages of standard PINNs, such as being meshfree and easy to set up.

Abstract

Temporally and spatially dependent uncertain parameters are regularly encountered in engineering applications. Commonly these uncertainties are accounted for using random fields and processes, which require knowledge about the appearing probability distributions functions that is not readily available. In these cases non-probabilistic approaches such as interval analysis and fuzzy set theory are helpful uncertainty measures. Partial differential equations involving fuzzy and interval fields are traditionally solved using the finite element method where the input fields are sampled using some basis function expansion methods. This approach however is problematic, as it is reliant on knowledge about the spatial correlation fields. In this work we utilize physics-informed neural networks (PINNs) to solve interval and fuzzy partial differential equations. The resulting network structures termed interval physics-informed neural networks (iPINNs) and fuzzy physics-informed neural networks (fPINNs) show promising results for obtaining bounded solutions of equations involving spatially and/or temporally uncertain parameter fields. In contrast to finite element approaches, no correlation length specification of the input fields as well as no Monte-Carlo simulations are necessary. In fact, information about the input interval fields is obtained directly as a byproduct of the presented solution scheme. Furthermore, all major advantages of PINNs are retained, i.e. meshfree nature of the scheme, and ease of inverse problem set-up.